A SIMPLIFIED SETTLING VELOCITY FORMULA FOR SEDIMENT PARTICLE By Nian-Sheng Cheng1
ABSTRACT: A new and simplified formula for predicting the settling velocity of natural sediment particles is developed. The formula proposes an explicit relationship between the particle Reynolds number and a dimensionless particle parameter. It is applicable to a wide range of Reynolds numbers from the Stokes flow to the turbulent regime. The proposed formula has the highest degree of prediction accuracy when compared with other published formulas. It also agrees well with the widely used diagrams and tables proposed by the U. S. Inter-Agency Committee (1957).

INTRODUCTION A pre-requisite to certain quantitative analysis in sediment transport is a knowledge of the settling velocity of sediment particles. Many attempts have been made for its prediction but most of the relevant researches apply only to spherical particles. Basically, there are two types of prediction methods for settling velocity of either spherical or non-spherical particles. One is the analytical solution of Stokes which is applicable only for particle Reynolds number, Re = wd/v ≤ 1, where w = settling velocity of a particle, d = particle diameter and v = kinematic viscosity. The other includes tabulated data and diagrams consisting of families of curves based on the experimental data, e.g. Schiller and Naumann (1933) and U. S. Inter-Agency Committee (1957). They are suited to a wide range of Reynolds numbers but inconvenient to use in practice. The objective of this note is to derive a simple expression for the determination of the settling velocity of natural sediment particles over a wide range Re. _____________________________________________________________________________ 1 School of Civil and Structural Engrg., Nanyang Technological University, Nanyang Avenue, Singapore, 639798 Key words: settling velocity, sediment particle, drag coefficient, Reynolds number

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DRAG COEFFICIENT FOR SETTLING OF INDIVIDUAL PARTICLE In 1851, Stokes obtained the solution for the drag resistance of flow past a sphere by expressing the simplified Navier-Stokes equation together with the continuity equation in polar co-ordinates. Using his solution, the following expression for the settling velocity of spherical particles can be derived as, 1 Δgd 2 (1) w= 18 ν where, Δ = (ρS - ρ)/ρ, ρ = density of fluid, ρS = density of particles, g = gravitational acceleration. Unfortunately, (1) is only valid for Re ≤ 1. Generally, by equating the effective weight force to the Newtonian expression of drag resistance, i.e. π π ρw 2 (2) ( ρ S − ρ )g d 3 = CD d 2 6 4 2 the drag coefficient CD can be expressed as 4 Δgd CD = (3) 3 w2 By substituting the Stokes’ solution in (1) into (3), CD may be related to the Reynolds number: A CD = (4) Re where A = a constant, which is dependent on the shape factor of the particle. For Stokes’ solution, A = 24 for spherical particles. The effect of particle shape on the drag coefficient varies, being small at low Re and more appreciable at high Re (Schulz, et al., 1954). Usually the shape factor of sediment particle is less than unity and for natural sand particles, the shape factor is about 0.7. Table 1 shows the value of A to be about 32 based on the work of various investigators. Under the condition of high Reynolds numbers, say, 103 ~ 105, the drag coefficient of spheres has an average value of about 0.4. For natural sediment particles, CD lies between 1.0 ~ 1.2 as shown in Table 1. As the Stokes-type equation is restricted to Re ≤ 1, efforts have been made to develop a method for extending (4) to a much wider range of Reynolds numbers. Some quasi-theoretical formulas or empirical correlations for evaluating the settling velocity of individual particle can be found in the literature, e.g., Oseen (1927), Sha (1956), Zanke (1977) and Raudkivi (1990). In the light of all these studies, the following relation between CD and Re is assumed for natural sediment particles: 1 1 A n CD = [( ) + B n ]n (5) Re where A and B are constants, and n = an exponent. Eq. (5) automatically satisfies the two extreme conditions at low and high Reynolds numbers, that is, CD is inversely proportional to Re at low Reynolds numbers and becomes a constant at high Reynolds numbers. According to Table 1, A can be taken as 32, as most researchers did, and B = 1, being the lowest limit of the drag coefficient for sediment particles. As the relationship between CD and Re at the extreme Reynolds numbers is unaffected appreciably by n in (5), the latter may be estimated by fitting (5) with the experimental data in the intermediate Reynolds number range, i.e., 1 < Re < 1000. Based on the experimental data of Concharov (1962: see Ibad-zade, 1992), Zegzhda (1934), Arkhangel’skii (1935) and Sarkisyan (1958) for quartz sand particles, the average n-value was found to be 1.5. Therefore, with the foregoing values proposed for A and B, (5) can be rewritten as 1 32 1.5 CD = [( ) + 1 ] 1.5 (6) Re
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Eq. (6) is a general relationship between the drag coefficient and particle Reynolds number for natural sediment particles. Using a dimensionless particle parameter d* defined as 1 Δg 3 d* = ( 2 ) d (7)

settling velocity of quartz sand particles in water at 20oC. Table 2 shows the 13 sets data points which are computed using the method outlined in Raudkivi (1990). The second is the experimental data of Zegzhda (1934), Arkhangel’skii (1935) and Sarkisyan (1958), which were compiled in the order of decreasing particle diameter by Zhu and Cheng (1993), as shown in Table 3. The last is the tabulated data given by the U. S. Inter-Agency Committee (1957) (also see: Raudkivi, 1990) for settling velocity of natural sediment particles with a shape factor of 0.7, and specific gravity ranging from 2.0 to 4.3. The basic parameter used for the determination of accuracy of a formula is the average value of the relative error defined as
error = | calculated − given| ×100 given

(15)

Table 2 presents the comparison of the calculated settling velocity using (9) together with those using the other five methods, with the average values reproduced from Raudkivi (1990). It can be seen that (9) has the smallest relative error when compared with the other formulas. The comparison given in Table 3 is between the various computed results and the experimental data of Zegzhda (1934), Arkhangel’skii (1935) and Sarkisyan (1958). It shows that the average relative error of (9) is 6.1%, which is very close to the 5.8% achieved by Zhu and Cheng’s (1993) formula and the degree of accuracy is better than all the other formulas. The present formula is also simpler to use than that proposed earlier by Zhu and Cheng (1993). Fig. 1 displays the relationship of Re and d* derived from (9) and it can be seen that the computed data also agree very well with the tabulated ones given by the U. S. Inter-Agency Committee (1957). CONCLUSIONS An explicit and simple formula was developed for evaluating the settling velocity of individual natural sediment particles. The formula is applicable to the different regimes ranging from the Stokes flow to the high Reynolds number. Comparison with published experimental data shows that the proposed formula has a high degree of prediction accuracy. ACKNOWLEDGEMENTS The author is thankful to Dr. Siow-Yong Lim and Dr. Yee-Meng Chiew, School of Civil and Structural Engineering, Nanyang Technological University, and the anonymous referees for their reviews and useful comments. APPENDIX I. REFERENCES 1. Arkhangel’skii, B. V. (1935). “Experimental study of accuracy of hydraulic coarseness scale of particles.” Izv. NIIG, No. 15, Moscow (in Russian). 2. Ibad-zade, Y. A. (1992). “Movement of sediment in open channels.” Translated by Ghosh, S. P. Russian Translations Series, Vol. 49. A. A. Balkema/Rotterdam. 3. Oseen, C. W. (1927). “Neuere Methoden und Ergebnisse in der Hydrodynamik.” Akademische Verlagsgesellschaft, Leipzig. 4. Raudkivi, A. J. (1990). “Loose boundary hydraulics.” 3rd edition, Pergamon Press. 5. Sarkisyan, A. A. (1958). “Deposition of sediment in a turbulent stream.” Izd. AN SSSR, Moscow (in Russian). 6. Schiller, L., and Naumann, A. (1933). “ Uber die grundlegenden Berechnungen bei der Schwekraftaubereitung.” Zeitschrift des Vereines Deutscher Ingenieure, 77(12), 318-320. 7. Schulz, S. F., Wilde, R. H., and Albertson, M. L. (1954). “Influence of shape on the fall velocity of sedimentary particles.” M. R. D. Sediment Series, No. 5, Missouri River Div., U. S. Corps of Engrs. 8. Sha, Y. Q. (1956). “Basic principles of sediment transport.” Journal of Sediment Research. 1(2): 1-54. (in Chinese). 9. U. S. Inter-Agency Committee (1957). “Some fundamentals of particle size analysis. A
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